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Stochastic Deformations of Sample Paths of Random Walks and Exclusion Models

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(1)Stochastic Deformations of Sample Paths of Random Walks and Exclusion Models Guy Fayolle, Cyril Furtlehner. To cite this version: Guy Fayolle, Cyril Furtlehner. Stochastic Deformations of Sample Paths of Random Walks and Exclusion Models. [Research Report] RR-5106, INRIA. 2004. �inria-00071476�. HAL Id: inria-00071476 https://hal.inria.fr/inria-00071476 Submitted on 23 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Stochastic Deformations of Sample Paths of Random Walks and Exclusion Models Guy Fayolle — Cyril Furtlehner. N° 5106 February 2004. ISSN 0249-6399. ISRN INRIA/RR--5106--FR+ENG. THÈME 1. apport de recherche.

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(151) ^S„. : 9&&( ] 

(152) & :  & ,. O·“–jnX³jnX p]]vMeUpnjn“”xSu”]ddnv]xS“”]d¬¢DjnX[]šžqBpn\ H({X}) =. ŸX[]Sp]3jnX[]3xqBy[dnj6eUy™jd. X i<j. αab , αbc , αca. αab = log. H. qUš+U @ ¥ ‚9VxqB\<]d

(153) jq. αab Ai Bj + αbc Bi Cj + αca Ci Aj ,. j6eU¡‡]jnX[]steUu–g[]d. p+ , p−. αbc = log. eUy[w~jnX[]xqBy[dnjnp6eU“–y™jd2U @ ¥ @ V˜y[qfŸ']xqB\<] αbc NA = ca , NB α.  $B.  G E  /. KL 0 M D

(154)   D I. UH„ ¥ 5^ V. q+ , q−. NB αca = ab , NC α. αca = log. r+ , r−. NC αab = bc . NA α. UH„ ¥ 9‚ V. G 4E FF  KJEMNG8D. y jnX[]]Oh eU\ v u”]qUšMjnX[]ÀÁ

(155) Â\<q—w ]Su Qã‚XT©¢tŸ˜]YXMes‡]

(156) eUjPXMeUy[w<eUy ]Ohmv u–“”xS“–jœeUyMeUu–lmjn“”x]Ohmv p]d6dn“”qBy. š›qBpªjnX[]“–y™steUpn“±eUy™j˜\<]feBdng p] ¥ P y š¨eBxSjf¢—qBg p˜xSu±eU“–\“”dœjnXMeUj˜]ɗgMeUjn“”qBy[dœ“–yŠjnX[]

(157) jnX[]Spn\<qmwDl—yMeU\ “”x u–“–\ “–j

(158) xfeUy³]w ]Spn“–s‡]wƁ™lig[dn“–y †<e \<]SjnX[q—wÆv pqBvqtd]wi“–y Q4S5T¾š›qBp˜jnX[]d6əgMeUp]u±eUjnjn“”x]\<qmw ]Su¨¢ ŸX “”x6X_ev pn“”qBpn“ /,5 +'    :  "  - 7$'&; 35' +&  / 7, qUšjnX[]˜“–y—sBeUpn“±eUy—j\<]feBdng p]t¢tŸX “”xX “”d\<qtdnj3qUš•jnX[]°jn“–\<]°g y™jnp6eBxSj6eU u”] ¥ WYX[] eUv v pq‡eBxXxqBy[dn“”djd“–y2xqBy[wD“–jn“”qBy “–y †ƒjnX[]°\<qBjn“”qBy qUše~j6eU†t†‡]w¤vMeUpnjn“”xSu”]_Ÿ“–jnX¤p]dnv]xSj&jq2eBwD\ “”ddn“– u”]ixqByD¶[†tg p6eUjn“”qBy[d ¥ O%]Šdn¡‡]Sjx6XºŸ“–jnX[qBg j v pqmqUšPjnX[]\_eU“–y~u–“–y[]d

(159) qUšªeUpn†tg \<]Sy™j“–y~jnX[]xfeBd]3qUšPjnX[]ÀÁ

(160) ·\<qmw ]Su ¥ MqBp£jnX[]_dReU¡‡]_qUškdnX[qBpnjny[]dd¬¢ (A, B, C) Ÿ“–u–u•w ]Sy[qBj]ŠjnX[] 3N ¦©wD“–\<]Sy[dn“”qByMeUuYqBu”]feUyís‡]xSjqBp {(Ai , Bi , Ci ), i = 1, . . . , N } ¥ ˜qBy[d“”w ]SpªjnX[]dnj]feBwDl dnj6eUj]Y]SXMesm“”qBg p•eBd t → ∞ eUy[w u”]Sj qa w ]Sy[qBj]

(161) jnX[]p6eUy[w qB\ steUpn“±eU u”] p]Sv p]d6]Sy™jn“–y †jnX[]œxqBy[wD“–jn“”qByMeUumv pqBMeU “–u–“–j©l£qUšm¶[y[wD“–y †evMeUpnjni“”xSu”] A qBy£dn“–j] i ¢U†t“–s‡]Sy (B, C) ¥ i P j“”dvqtddn“– u”]t¢[eUu–jnX[qBg †tXƒwD“§­_xSg u–jf¢Mjq<dnX[q¬ŸjnXMeUj P. P (A | B, C) =. N Y  i=1. ŸX[]Sp]3jnX[]3xqBy[wD“–jn“”qByMeUu4]ɗg “–u–“– pn“–g \ qUšPj©lmv].  Ai qia + Ai (1 − qia ) , A. vMeUpnjn“”xSu”]dkd6eUjn“”dn¶M]d. a qi+1 λ+ (i) = −a , a qi λa (i + 1). ÒHӌÔKҞÕ.

(162) 

(163) *-   *!% %$'&  )+ 99 "! .+ /9. 0 &?3 '/ 6 79 &;:,  < /, &4. ^f{. Ÿ“–jnX)xqBy[wD“–jn“”qByMeUuKp6eUj]d†t“–s‡]Sy~—l  + − λ+ a (i) = p Bi + r Ci + Γ B i C i ,. λ− (i + 1) = p− B + r + C + Γ B C , i i i i a. jnX[]d6]ƒ]OhDv p]ddn“”qBy[dŠ]S“–y †¤dn“–\ “–u±eUpŠjqjnX[]~š›qBpn\ qUš U›‚ ¥ 9‚ V<“–y¸jnX[] \<qmw ]Su ¥ “”dÆe y[qByD¦¨µf]SpqÆɗgMeUy™jn“–j lƒj6eU¡m“–y †_“–y™jq³eBxxqBg y—jjnX[] ]Oh xSu–g[dn“”qBy2eUjdn“–j] i “–{τy¯ajnτX[b]3} š›qBu–u”qfŸ“–y †³Γd]Sy[d6]t¿ ŸX[]Sy B C = 1 jnX[]Sy2y[]x]dd6eUpn“–u–l A = 1 ¥ WYX[] v pqmqUšœp]Su–“”]dqBy)jnX[] xqBy™s‡]Spn†‡]Sy[x]<qUš•eUy i “–j]Sp6eUjn“–s‡]3i pq—i x6¡—“–y †B¦©dx6X[]S\<] ¥ Á˜lijnX[]qmqBu”]feUy~yMeUjng p]3qUš B eUy[w C ¢ xfeUy~Ÿpn“–j] i. i. log. Ÿ“–jnXƒj©Ÿ•qŠeUyMeUu”qB†‡qBg[d]ɗgMeUjn“”qBy[dš›qBp B eUy[w C dnv]xS“”]d ¥

(164) 4Ñ4  Ì f ˜‹f‘‡  O%]“–y—jnpq—wDg[x]£jnX[]3dqU¦©xfeUu–u”]w αbc =. 1 α +o , N N. αca =. : '/9 R  &#  &;7. 1 β +o , N N. αab =. ŸX[]Sp] α, β, γ eUp]jnX p]]vqtd“–jn“–s‡]p]feUuKxqBy[dnj6eUy—j ¥   F  Ì

(165) KD Ð 4‹ « ]Sjnjn“–y †ÆšžpqB\Äy[q¬Ÿ qBy 

(166) . ¢šžqBp x = i/N jnXMeUjkjnX[]Ÿ•]feU¡Æu–“–\ “–jd a ρa (x) = lim qxN , N →∞. UH„ ¥ @ V. a qi+1 = αca Ci − αab Bi , qia. b ρb (x) = lim qxN , N →∞. w ]O¶[y[]w~—l. 1 γ +o , N N. 1≤i≤N. ¢¾qBy[]°xfeUy?dnX[q¬Ÿ. c ρc (x) = lim qxN N →∞. ]OhD“”dnjkeUy[w~dReUjn“”d»šHlŠjnX[]dnlDdnj]S\ qUšPw ]Sj]Spn\ “–y “”djn“”x£wD“§½¾]Sp]Sy™jn“±eUu4]ɗgMeUjn“”qBy[d  ∂ρ   a = ρa (βρc − γρb ),     ∂x ∂ρb = ρb (γρa − αρc ),  ∂x      ∂ρc = ρc (αρb − βρa ), ∂x. UH„. ¥„. Ÿ“–jnXƒjnX[]xSpng[xS“±eUu4xqBy[dnjnp6eU“–y—jdkwDg[]jq jnX[]v]Spn“”qmwD“”xqBg y[w[eUpnl³xqBy[wD“–jn“”qBy[d ρu (x + 1) = ρu (x),. Ô4Ô Ü^[ïfì»î_. ∀u = a, b, c.. UH„ ¥ 9{ V. V.

(167) ^f}. : 9&&( ] 

(168) & :  & ,.  « j6eUpnjn“–y †išžpqB\ UH„ ¥ @ VO¢¾qBy[]°eUv v u–“”]djnX[]£u±efŸ qUš•u±eUpn†‡]£y—g \£]Spd3eUy[w¯]Spn†‡qmwD“”x&jnX[]qU¦ p]S\<d

(169) jqråg[djn“§šHlŠjnX[]£eUv v pqhD“–\_eUjn“”qBy)qUšŒ¶[y “–j]3dng \<d—l³a

(170) “”]S\_eUy y~“–y—j]S†tp6eUu”df¢eBd N → ∞ ¥ !&-&%. j<“”dieU\£g[dn“–y †¤jqd]]~jnXMeUj UH„ ¥ „ V°]Su”qBy †‡dŠjq%jnX[]ƒxSu±eBd6d_qUš†‡]Sy[]Sp6eUu–“–µf]w ´ qBjn¡Be¦¨ÅªqBu–j]Spnp6e d lmdnj]S\<d ¥ WYX[]

(171) qBpn“–†t“–yMeUu ´ qBjn¡te¦¨ÅªqBu–j]Spnp6e\<qmw ]Su[Ÿ˜eBdœjnX[]

(172) dn“–\ v u”]dnjª\<qmw ]SuMqUšv p]w[eUjqBp»¦¨v p]Sl “–y—j]Sp6eBxSjn“”qBy[df¢v pqBvqtd]w%“–y[w ]Sv]Sy[w ]Sy™jnu–l큙l ´ qBjn¡Be U»^ ,t‚t{9V£eUy[wŪqBu–j]Spnp6e U»^ ,t‚t}9VO¢d]] š›qBp “–y[dj6eUy[x] Q–^ @ T ¥ qBy[]SjnX[]Su”]ddf¢“–y?qBg pxfeBd6]£jnX[] Ÿ•qBpnu”w¯“”du”]ddxSpngMeUuªeUy[w¯eUu–uPvMeUpnjn“”xSu”] j©lmv]d eUp]qBy¯eUyƒ]əgMeUuš›qmqBjn“–y † ¥–¥–¥ WYX[] šžqBpn\ qUš UH„ ¥ „ Vu”]Sy[w d£“–jd]Su§š˜jq)eÆdqBu–g jn“”qBy“–y?j]Spn\<d&qUšYdnv]xS“±eUušžg y[xSjn“”qBy[d ¥ WYX[] ¶[pdnj dj]Sv³“”djq p]S\_eUpn¡ijnX[]3]Ohm“”dj]Sy[x]3qUšŒj Ÿ•q_u”]Ss‡]SuAdng p»š›eBx]d P. . UH„ ¥ }9V UH„ ¥ SV. ρa + ρb + ρc = 1,. ŸX[]Sp]UH„  dn“–y † HU „. ραa ρβb ργc. ¥ }9VAš›qBu–u”qfŸ

(173) dªeUjPqBy[x]œšžpqB\ UH„ ¥ 9‚ VŒeUy[w κ ”“ dexqBy[dnj6eUy™jªqUšM\<qBjn“”qBy°jq]•w ]Sj]Spn\ “–y[]w ¥ ¥ 9} V˜jq ]Su–“–\ “–yMeUj] ρc ¢DŸ˜]p]SŸpn“–j] UH„ ¥ SVBe d UH„ ¥ V ρα ρβ (1 − ρ − ρ )γ = κ.. WYX[]˜xXMeUy †‡]YqUš šžg y[xSjn“”qBy[d wD“§½¾]Sp]Sy™jn“±eUuŒ]əgMeUjn“”qBy ŸX[]Sp]. = κ,. v(u). a b. u = ρa +ρb. d6eUjn“”dn¶M]djnX[]3]ɗgMeUjn“”qBy. eUy[w. a. b. v = βρa −αρb. ¢lm“”]Su”w d4jnX[]ª¶[pdnjŒqBpw ]Sp4y[qBy u–“–y[]feUp. (αu + v)α (βu − v)β (1 − u)γ = κ(α + β)α+β. MqBpn\_eUu–u–l‡¢. u(x). UH„ ¥ ,9V. du = (1 − u)v(u) dx. UH„. ¥ ^f|9V. UH„. ¥ ^t^5V. xfeUy³]]OhDv p]dd]wƒeBd x=. Z. u(x). u( 0). du . (1 − u)v(u). j eUv v]feUpd£jnXMeUj&jnX[]<p6eUjn“”qtd “–y UH„ ¥ ‚9V&eUp]<p6eUjn“”qByMeUu•šžqBp eUu–uª¶[y “–j] N ¢4 g jf¢Œd“–y[x]<Ÿ•]ŠXMes‡] & u”]Sj N, N , N , → ∞ ¢™jnX[]SlŠ\ “–†tX™j]xqB\<]eUpn “–j6eUpnlŠp]feUuy™g \&]SpdY“–yÆjnX[]“–y™j]SpnsteUu [0, 1] ¥ O·X[]Sy¤jnX[A]SleUBp]_p6eUjn“”qByMeUu¨\ ¢ UH„ ¥ ,9V3“”d°e~vqBu–l—y[qB\ “±eUuY]ɗgMeUjn“”qByA¢ªeUy[wjnX[]Sy u(x) “”d&“–y¤dqB\<] P. ÒHӌÔKҞÕ.

(174) 0 &?3 '/ 6 79 &;:,  < /, &4 5^ S. 

(175) *-   *!% %$'&  )+ 99 "! .+ /9. d6]Sy[d]4råg[dnjªe “–j\<qBp]Y†‡]Sy[]Sp6eUu jnXMeUjœeUy°X™lmv]Sp]Su–u–“–v jn“”xšHg y[xSjn“”qByA¢—dn“–y[x]t¢™ešžj]SpPd6qB\<]YeUu–†‡]S p6eD¢ Ÿ˜]£eUp]u”]OšžjŸ“–jnX~“–y™j]S†tp6eUu”dqUšŒjnX[]šžqBpn\ Z q  −p  s2 − a2 (1 − s) q ds,. ŸX[]Sp] p eUy[w q dnj6eUy[wƒšžqBpvqtdn“–jn“–s‡]&“–y™j]S†‡]Spd ¥ WYX[] vMeUpnjn“”xSg u±eUp3xfeBd] d“–\ v u”]t¢[dn“–y[x]jnX[]Sy . du dx. 2. α=β=γ. “”dp6eUjnX[]Sp. = (1 − u)[u2 (1 − u) − κ],. dX[qfŸ“–y † u jq<]3e°dnj6eUy[w[eUpw™eBxqB “K]Su–u–“–v jn“”xšžg y[xSjn“”qBy ¥ MqBpYeUy—l_†t“–s‡]Sy κ ¢mjnX[]dnlDdnj]S\ UH„ ¥ „ VœeBwD\ “–jdqUš4e£g y “”É—g[]dqBu–g jn“”qBy ¥ ©P yŠvMeUpnjn“”xSg u±eUpf¢DjnX[]Sp]“”d & K 

(176) e w ]S†‡]Sy[]Sp6eUj]&dqBu–g jn“”qByƒxSpng \£ u”]wƒjq°jnX[]¶ h ]w³vqB“–y—j ρea =. α , α+β +γ. eUy[wƒxqBpnp]dnvqBy[wD“–y †_jq jnX[]xqBy[dnj6eUy™j. ρeb =. β , α+β+γ. ρec =. γ , α+β +γ. UH„. ¥ ^f‚9V. ¥^ @ V WYX[]v g pnvqtd]qUš4jnX[]y[]OhDj˜vMeUp6eU†tp6eUv Xƒ“”d•jq wD“”d6xSpn“–\ “–yMeUj]]Sj Ÿ•]]SyƒdqBu–g jn“”qBy[dqUš+UH„ ¥ „ V˜eUy[w ¢[eBd N → ∞ ¥ UH„ ¥ {9VO¢[“–y~qBpw ]Sp

(177) jq p]Su±eUj]jnX[]S\ jqŠeBwD\ “”ddn“– u”]u–“–\ “–j»¦¨vqB“–y™jdqUš ΦN κ e  =.  $.. αα β β γ γ . (α + β + γ)α+β+γ. UH„. 8E   M M?E4  JAD I0!" DE 0 % FM;G8I  D#I     EF  D,M?EMNG8D. 4W qŠxfeUjx6X2e<pqBg †tX¯É—gMeUu–“–j6eUjn“–s‡]&“–y[dn“–†tX™j“–y™jqŠjnX[]&d6qBu–g jn“”qBy¯qUš\UH„ ¥ „ VO¢¾e<dnj6eUy[w[eUpw2eUv v pq‡eBxX p]Su–“”]d&qByíeÆu–“–y[]feUpn“–µ¬eUjn“”qByºqUš˜jnX[]_pn“–†tX™j»¦¨XMeUy[w¤dn“”w ]_eUpqBg y[wjnX[]°¶ hD]w%vqB“–y—j UH„ ¥ ^f‚9V ¥ WYX “”d lm“”]Su”w dke u–“–y[]feUpkwD“§½¾]Sp]Sy™jn“±eUuŒdnlDdnj]S\~¢DŸX[qtd]\_eUjnpn“§h .  0 −αγ αβ 1  βγ 0 −αβ  , α+β+γ −βγ αγ 0 p eUy[w λ = ±i αβγ(α + β + γ) ¢keUy[w·jnX[])jnp6er»]xSjqBpn“”]d~eUp] 0 ±. XMeBdijnX p]]2]S“–†‡]Sy™steUu–g[]d u”qmxfeUj]wƒqBy?eUy)]Su–u–“–v[d6qB“”w ¥ Á˜g jf¢dn“–y[x]£jnX[] eUqfs‡] ]S“–†‡]Sy™steUu–g[]deUp]3v g p]Su–l~“–\_eU†t“–yMeUpnl‡¢“–j“”d Ÿ˜]Su–u¡—y[q¬ŸyijnXMeUjYy[q°xqBy[xSu–g[dn“”qBy³xfeUyƁ]wDp6eŸyƒeBd•š›qBp•jnX[]qBpn“–†t“–yMeUu¾dnlDdnj]S\~¢mŸX “”xXÆ\ “–†tX™j ]qUšªe°É™g “–j]3wD“§½¾]Sp]Sy™jkyMeUjng p] ¥ 8qfŸ˜]Ss‡]Spf¢“–j“”dkv u”]feBd6eUy—jjqŠd6]]£jnXMeUjjnX[]&\<q—wDg u–g[dqUšœjnX[] y[qBy µf]SpqÆ]S“–†‡]Sy™steUu–g[]dv u±elDd“–y š¨eBxSjke°xSpng[xS“±eUuKpqBu”]&eBd

(178) dnX[qfŸyƒ“–y³jnX[]y[]OhmjjnX[]qBp]S\ ¥ Ô4Ô Ü^[ïfì»î_.

(179) ^. : 9&&( ] 

(180) & :  & ,.  . '/ 

(181)  s  5  &; 2 

(182) . !  Z\[-]  )  &9!  9   

(183) . s = α+β+γ η = Φ = lim ΦN / &\ /,   2   '/   > =   '3/  /,R$ 

(184) > N →∞  $  ' ' 5  2 7 -   & 

(185) &?: . . 2π ηc 

(186) = p , 3 ρea ρeb ρec :                   &(

(187)  / ' !/, ,7             3  4  7  3  η8 > η$c / T (κ ) = 1  p ∈ {1, . . . , [ η ]}  5     & 9/   &(2  &  Φ p p ηc. . 

(188) .  . . . . .    $ '/

(189) . .     -!   

(190)      / 8. κ1. . η > ηc. 

(191)   2  3.    /, 7,       7 !   7&  $ '. . . η ≤ ηc. . WYX[]Æv pqmqUš“–y™s‡qBu–s‡]dŠe)š›qBp]dnj<qUšj]xX y “”xfeUu–“–jn“”]dÆeUy[wºŸ•]~qBy u–lºdn¡‡]Sjx6X jnX[]Æ\_eU“–y u–“–y[]d<qUš eUpn†tg \<]Sy—j ¥ WYX[]¶[pdj

(192) dnj]Sv~“”djq dnŸ“–jx6X~jq vqBu±eUp

(193) xq—qBpwD“–yMeUj]d  α ua 

(194) = ρea − = r cos θ s u 

(195) = ρe − α = r sin θ. b b s. a]SŸpn“–j]UH„ ¥ SVeBd. UH„. H(r, θ) = log κ,. Ÿ“–jnX. ¥ ^S„. V.   γ  α β H(r, θ) 

(196) = α log r cos θ + + β log r sin θ + + γ log − r(cos θ + sin θ) , s s s. eUy[w¸u”]Sj ]³jnX[])dn“–y †tu”]~pqmqBj_“–y ]ɗgMeUjn“”qBy r(θ, κ). r. qUš HU „ ¥ ^S„ V ¥ WYX[]Sy. θ. d6eUjn“”dn¶M]d_jnX[])wD“§½]Sp]Sy—jn“±eUu UH„. dθ = G(θ, κ), dx. ŸX[]Sp]. ¥ ^f{9V. G(θ, κ)  =. ´ ]Sjnjn“–y †.  1 β(α + γ) cos θ + αβ sin2 θ + α(β + γ) sin2 θ s   + r(θ, κ) cos θ sin θ (β + 2α + 2γ) cos θ + (α + 2β + 2γ) sin θ .. T (κ). ]jnX[]v]Spn“”q—w³qUšPjnX[]3qBpn “–jf¢ Ÿ˜]XMes‡] T (κ) =. Z. 2π 0. dθ . G(θ, κ). ÒHӌÔKҞÕ.

(197) 

(198) *-   *!% %$'&  )+ 99 "! .+ /9. 0 &?3 '/ 6 79 &;:,  < /, &4 ^ ,. WYX[] d]xqBy[w)“–\ vqBpnj6eUy—j3dnj]Sv¯p]Su–“”]dqBy¯jnX[]&\<qBy[qBjqBy “”x&]SXMesm“”qBg pqUš jq°jnX[]3vMeUp6eU\<]Sj]Sp κ ¢ lm“”]Su”wD“–y †<jnX[]“–y[]ɗgMeUu–“–j©l #. Ÿ“–jnX2p]dnv]xSj. T (κ) ≥ T (e κ).. [d6]Spns—“–y †<jnXMeUj. Ÿ˜]3xfeUy~Ÿpn“–j]t¢ —l>UH„ ¥ ^f{9VO¢ T (e κ) = −4is. T (κ). I. Γ. r(θ, κ e) = 0,. T (e κ). z 2 [γ(β. ∀ θ ∈ [0, 2π],. eBdke°xqBy—jqBg pk“–y™j]S†tp6eUuŒqBy³jnX[]g y “–jkxS“–pxSu”]t¢ yMeU\<]Su–l. 1 , − α) − 2iαβ] + 2z[2αβ + γ(α + β)] + γ(β − α) + 2iαβ. qBpf¢[ešžj]Spe°dn“–\ v u”]£xfeUu”xSg u–g[df¢. T (e κ) = 2π. ŸX “”x6X~u”]feBw d

(199) v p]xS“”d6]Su–lijq°jnX[]£xSpn“–jn“”xfeUuAsBeUu–g[]. r. ηc. s , αβγ. eUy y[qBg y[x]w~“–y³jnX[]jnX[]qBp]S\ ¥. &+ F' L EM . WYX “”dvMeUv]Spk“”djnX[]&xqBy—jn“–y™gMeUjn“”qBy%qUš Q4SXT©¢[ g j“”dx]Spnj6eU“–y u–l)eUyƒ“–y—j]Spn\<]wD“±eUj] dnj]Sv ¥ [qBpjnX[] dReU¡‡]•qUšdX[qBpnjny[]ddf¢UŸ•]YwD“”w&p]dnjnpn“”xSjqBg pd]Su–s‡]dŒjqjnX[]•jnX[]Spn\<qmwDl—yMeU\ “”xdqUšMjnX[]˜ÀÁÂ%\<qmw ]Su ¥ ÀxSjngMeUu–u–l‡¢qBg p†‡q‡eUuŒ“”dkjqÆeUyMeUu–l—µf]&jnX[]°wDl—yMeU\ “”xdqUšªp6eUy[w qB\ xSg pns‡]d]Ss‡qBu–s—“–y †Æ“–y Zm UHy[q dvMeUjn“±eUuªxqBy[dnjnp6eU“–y™jd VqBp“–y Zm ¢¾ŸX[]Sy?jnX[]Sl¯Ÿ˜eUpnv%g y[w ]SpjnX[]<eBxSjn“”qByíqUš˜dqB\<] dnjqmxXMeBdjn“”x + w ]Oš›qBpn\_eUjn“”qBy~†tp6eU\ \_eUp ¥ P y QãX} T©¢jnX “”dv pqUrn]xSjŸ“–u–uP] xfeUpnpn“”]w?qBg j“–y)jnX[]&šžp6eU\<]SŸ•qBpn¡2qUšœu±eUpn†‡]°dxfeUu”]°wDl—yMeU\ “”xdš›qBp ]Oh xSu–g[dn“”qBy~v pqmx]dd]d¬¢DeUy[w~“–j

(200) Ÿ“–u–u¾\_eU“–y u–l~eBw wDp]dd

(201) jnX[]vqB“–y™jdku–“”dnj]w³X[]Sp]fešžj]Sp ¥ *Ð Ž ÌiÐ í  Ì Ž  Ð Ñ Í 4 H‘ Ì

(202)  H‰4Ž 

(203)  WYX[]kjnpn“”x6¡<jq£w ]Spn“–s‡]u–“–\ “–jn“–y † wD“§½¾]Sp]Sy™jn“±eUu dlmdnj]S\<d•eU\<qBg y™jd˜]dd]Sy™jn“±eUu–u–l jq3Ÿpn“–jn“–y † xqBy[wD“–jn“”qByMeUuMÇMqfŸ¼]əgMeUjn“”qBy[d•qBy_dng “–j6eU u”]kdReU\ v u”] vMeUjnX[d¬¢ ]Ss‡]Sy³“–y³jnX[]v p]d6]Sy[x]3qUš4vMeUpnjn“”xSu”]3xSg pnp]Sy—jd ¥ WYX[]d]]ɗgMeUjn“”qBy[d“–y—s‡qBu–s‡]šHg y[xSjn“”qByMeUu”d qUš®2eUpn¡‡qfs%eUy[w2jnX[]Sl?]SyBr»q¬l¯dnv]xS“±eUuš›]feUjng p]d3]Sy[xqBg y™j]Sp]w퓖y%\_eUy—l2dlmdnj]S\<d ¥ P j\ “–†tX™j eUu”d6q

(204) ]ª“–y™j]Sp]djn“–y †jqy[qBj]•jnXMeUjK\<qtdnjPqUš[jnX[] ´ qBjn¡Be¦¨ÅªqBu–j]Spnp6e]ɗgMeUjn“”qBy[dŒxfeUy ]œ]OhDv u±eU“–y[]w “–y³jnX[]3u–“–†tX™jkqUšPjnX[]š¨eU\<qBg[dg pny[dkqUš  X p]SyDš›]dnj ¥  K ‹ Ì fŽ 4‹ F  Ð  WYX[]Sp] ]OhD“”dnjde_†tu”qBMeUu“–y™j]Spnv p]Sj6eUjn“”qBy%—l~\<]feUy[dqUš˜e_šHp]] ]Sy[]Spn†tl šžg y[xSjn“”qByMeUuŒŸ“–jnX)j Ÿ•qÆxqB\ vqBy[]Sy—jdf¿YjnX[] ]Sy™jnpqBv™l¯qUšªjnX[]&dlmdnj]S\~¢eUy[wƒjnX[] eUu–†‡]S p6eU“”x eUp]fe ]Sy[xSu”qtd6]w_—l<jnX[]xSg pns‡] ¥ P jœjng pny[dYqBg j˜jnXMeUj•jnX[]xqBy™j]Sy—jn“”qBy³]Sj Ÿ•]]SyÆjnX[]d]j©Ÿ˜q ɗgMeUy™jn“–jn“”]d. Ô4Ô Ü^[ïfì»î_.

(205) ‚t|. : 9&&( ] 

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